Question: Determine the value of the following complex number power. Your answer will be plotted in orange. $ ({\cos(\frac{11}{12}\pi) + i \sin(\frac{11}{12}\pi)}) ^ {4} $
Let's express our complex number in Euler form first. $ {\cos(\frac{11}{12}\pi) + i \sin(\frac{11}{12}\pi)} = { e^{11\pi i / 12}} $ Since $(a ^ b) ^ c = a ^ {b \cdot c}$ $ ({ e^{11\pi i / 12}}) ^ {4} = e ^ {4 \cdot (11\pi i / 12)} $ The angle of the result is $4 \cdot \frac{11}{12}\pi$ , which is $\frac{11}{3}\pi$ Our result is $ e^{5\pi i / 3}$. Converting this back from Euler form, we get $\cos(\frac{5}{3}\pi) + i \sin(\frac{5}{3}\pi)$.